Monoamine oxidases (MAOs) catalyse the oxidation of neurotransmitter amines and a wide variety of primary, secondary and tertiary amine xenobiotics, including therapeutic drugs. While inhibition of MAO activity in the periphery removes protection from biogenic amines and so is undesirable, inhibition in the brain gives vital antidepressant and behavioural advantages that make MAO a major pharmaceutical target for inhibitor design. In neurodegenerative diseases, MAO inhibitors can help to maintain neurotransmitter levels, making it a common feature in novel multi-target combinations designed to combat Alzheimer's disease, albeit not yet proven clinically. Vital information for inhibitor design comes from an understanding of the structure, mechanism, and kinetics of the catalyst. This review will summarize the kinetic behaviour of MAO A and B and the kinetic evaluation of reversible inhibitors that transiently decrease catalysis. Kinetic parameters and crystal structures have enabled computational approaches to ligand discovery and validation of hits by docking. Kinetics and a wide variety of substrates and inhibitors along with theoretical modelling have also contributed to proposed schemes for the still debated chemical mechanism of amine oxidation. However, most of the marketed MAO drugs are long-lasting irreversible inactivators. The mechanism of irreversible inhibition by hydrazine, cyclopropylamine, and propargylamine drugs will be discussed. The article finishes with some examples of the propargylamine moiety in multi-target ligand design to combat neurodegeneration.